All right, so it was time I attacked this problem.
At first I scratched as must of the rest as possible with a sand paper "brick" and a scratcher.
Then I treated the whole surface with a special liquid that is supposed to stabilize rust which turned the whole basin dark blue.
Then I got this paint that not only can be applied on top of rust but is meant to treat it and protect the metal from further corrosion. Plus it came in a "aluminum" silvery gray color which looks pretty close to the original bare metal. I treated the whole inside of the basin along with the top edges. It needs to coats and I am doing the second one tonight, 24 hours after the first one. I am considering another coat just to be safe but the paint is expensive and I am not sure if the two cans I got would be enough for three applications.
The only problem is that it is applied by brush and let's say, my brush work would be fine for a painter but not for a wall painter; really messy looking.
I am also thinking of covering the whole inside with a sticky plastic sheet, making sure the edges have no... edges, but it seems to be mighty difficult to cut the sheet right then apply it without any bumps or imperfections. Another idea would be a thicker sort of plastic, like the PVC you mentioned, cut to size and then cover the edges with a silicone gun (which I happen to have). The easiest of all would be of course to just lay a plastic sheet on top, but that would look really messy and ugly.